Algebraically closed group

Algebraically closed group

In mathematics, in the realm of group theory, a group A is algebraically closed if any finite set of equations and inequations that "make sense" in A already have a solution in A . This idea will be made precise later in the article.

Informal discussion

Suppose we wished to find an element x of a group G satisfying the conditions (equations and inequations):

::x^2=1 ::x^3=1 ::x e 1

Then it is easy to see that this is impossible because the first two equations imply x=1 . In this case we say the set of conditions are inconsistent with G . (In fact this set of conditions are inconsistent with any group whatsoever.)

However if we extend the group G to the group H with multiplication table:

Then the condions have two solutions, namely x=b and x=c .

Thus there are three possibilities regarding such conditions:
* They may be inconsistent with G and have no solution in any extension of G .
* They may have a solution in G .
* They may have no solution in G but nevertheless have a solution in some extension H of G .

It is reasonable to ask whether there are any groups A such that whenever a set of conditions like these have a solution at all, they have a solution in A itself? The answer turns out to be "yes", and we call such groups algebraically closed groups.

Formal definition of an algebraically closed group

We first need some preliminary ideas.

If G is a group and F is the free group on countably many generators, then by a finite set of equations and inequations with coefficients in G we mean a pair of subsets E and I of Fstar G the free product of F and G .

This formalizes the notion of a set of equations and inequations consisting of variables x_i and elements g_j of G . The set E represents equations like: ::x_1^2g_1^4x_3=1::x_3^2g_2x_4g_1=1::dots The set I represents inequations like::g_5^{-1}x_3 e 1::dots

By a solution in G to this finite set of equations and inequations, we mean a homomorphism f:F ightarrow G, such that ilde{f}(e)=1 for all ein E and ilde{f}(i) e 1 for all iin I. Where ilde{f} is the unique homomorphism ilde{f}:Fstar G ightarrow G that equals f on F and is the identity on G .

This formalizes the idea of substituting elements of G for the variables to get true identities and inidentities. In the example the substitutions x_1mapsto g_6, x_3mapsto g_7 and x_4mapsto g_8 yield:::g_6^2g_1^4g_7=1::g_7^2g_2g_8g_1=1::dots ::g_5^{-1}g_7 e 1::dots

We say the finite set of equations and inequations is consistent with G if we can solve them in a "bigger" group H . More formally:

The equations and inequations are consistent with G if there is a groupH and an embedding h:G ightarrow H such that the finite set of equations and inequations ilde{h}(E) and ilde{h}(I) has a solution in H . Where ilde{h} is the unique homomorphism ilde{h}:Fstar G ightarrow Fstar H that equals h on G and is the identity on F .

Now we formally define the group A to be algebraically closed if every finite set of equations and inequations that has coefficients in A and is consistent with A has a solution in A .

Known Results

It is difficult to give concrete examples of algebraically closed groups as the following results indicate:

* Every countable group can be embedded in a countable algebraically closed group.
* Every algebraically closed group is simple.
* No algebraically closed group is finitely generated.
* An algebraically closed group cannot be recursively presented.
* A finitely generated group has solvable word problem if and only if it can embedded in every algebraically closed group.

The proofs of these results are, in general very complex. However a sketch the proof that a countable group C can be embedded in an algebraically closed group follows.

First we embed C in a countable group C_1 with the property that every finite set of equations with coefficients in C that is consistent in C_1 has a solution in C_1 as follows:

There are only countably many finite sets of equations and inequations with coefficients in C . Fix an enumeration S_0,S_1,S_2,dots of them. Define groups D_0,D_1,D_2,dots inductively by:

::D_0 = C

::D_{i+1} = left{egin{matrix} D_i &mbox{if} S_i mbox{is not consistent with} D_i \langle D_i,h_1,h_2,dots,h_n angle &mbox{if} S_i mbox{has a solution in} Hsupseteq D_i mbox{with} x_jmapsto h_j 1le jle nend{matrix} ight.

Now let:


Now iterate this construction to get a sequence of groups C=C_0,C_1,C_2,dots and let:


Then A is a countable group containing C . It is algebraically closed because any finite set of equations and inequations that is consistent with A must have coefficients in some C_i and so must have a solution in C_{i+1} .


* A. Macintyre: On algebraically closed groups, ann. of Math, 96, 53-97 (1972)
* B.H. Neumann: A note on algebraically closed groups. J. London Math. Soc. 27, 227-242 (1952)
* B.H. Neumann: The isomorphism problem for algebraically closed groups. In: Word Problems, pp 553-562. Amsterdam: North-Holland 1973
* W.R. Scott: Algebraically closed groups. Proc. Amer. Math. Soc. 2, 118-121 (1951)

Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Quasi-algebraically closed field — In mathematics, a field F is called quasi algebraically closed (or C1) if for every non constant homogeneous polynomial P over F has a non trivial zero provided the number of its variables is more than its degree. In other words, if P is a non… …   Wikipedia

  • Pseudo algebraically closed field — In mathematics, a field K is pseudo algebraically closed (usually abbreviated by PAC) if one of the following equivalent conditions holds:*Each absolutely irreducible variety V defined over K has a K rational point. *Each absolutely irreducible… …   Wikipedia

  • Group representation — In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group… …   Wikipedia

  • Absolute presentation of a group — In mathematics, one method of defining a group is by an absolute presentation.B. Neumann, The isomorphism problem for algebraically closed groups, in: Word Problems, Decision Problems, and the Burnside Problem in Group Theory, Amsterdam London… …   Wikipedia

  • Stable group — For stable groups in homotopy theory see stable homotopy group or direct limit of groups. In model theory, a stable group is a group that is stable in the sense of stability theory. An important class of examples is provided by groups of finite… …   Wikipedia

  • Real closed field — In mathematics, a real closed field is a field F in which any of the following equivalent conditions are true:#There is a total order on F making it an ordered field such that, in this ordering, every positive element of F is a square in F and… …   Wikipedia

  • Absolute Galois group — In mathematics, the absolute Galois group GK of a field K is the Galois group of K sep over K , where K sep is a separable closure of K . Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K . The absolute… …   Wikipedia

  • Formal group — In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group. They were first defined in 1946 by S. Bochner. The term formal group sometimes means the same as formal group law,… …   Wikipedia

  • Brauer group — In mathematics, the Brauer group arose out of an attempt to classify division algebras over a given field K . It is an abelian group with elements isomorphism classes of division algebras over K , such that the center is exactly K . The group is… …   Wikipedia

  • Existentially closed model — In model theory, a branch of mathematical logic, the notion of an existentially closed model of a theory generalizes the notions of algebraically closed fields (for the theory of fields), real closed fields (for the theory of ordered fields),… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”